{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CONTINUOUS\n",
      "['alpha', 'anglit', 'arcsine', 'beta', 'betaprime', 'bradford', 'burr', 'cauchy', 'chi', 'chi2', 'cosine', 'dgamma', 'dweibull', 'erlang', 'expon', 'exponnorm', 'exponpow', 'exponweib', 'f', 'fatiguelife', 'fisk', 'foldcauchy', 'foldnorm', 'frechet_l', 'frechet_r', 'gamma', 'gausshyper', 'genexpon', 'genextreme', 'gengamma', 'genhalflogistic', 'genlogistic', 'gennorm', 'genpareto', 'gilbrat', 'gompertz', 'gumbel_l', 'gumbel_r', 'halfcauchy', 'halfgennorm', 'halflogistic', 'halfnorm', 'hypsecant', 'invgamma', 'invgauss', 'invweibull', 'johnsonsb', 'johnsonsu', 'ksone', 'kstwobign', 'laplace', 'levy', 'levy_l', 'levy_stable', 'loggamma', 'logistic', 'loglaplace', 'lognorm', 'lomax', 'maxwell', 'mielke', 'nakagami', 'ncf', 'nct', 'ncx2', 'norm', 'pareto', 'pearson3', 'powerlaw', 'powerlognorm', 'powernorm', 'rayleigh', 'rdist', 'recipinvgauss', 'reciprocal', 'rice', 'semicircular', 't', 'triang', 'truncexpon', 'truncnorm', 'tukeylambda', 'uniform', 'vonmises', 'vonmises_line', 'wald', 'weibull_max', 'weibull_min', 'wrapcauchy']\n",
      "\n",
      "DISCRETE\n",
      "['bernoulli', 'binom', 'boltzmann', 'dlaplace', 'geom', 'hypergeom', 'logser', 'nbinom', 'planck', 'poisson', 'randint', 'skellam', 'zipf']\n"
     ]
    }
   ],
   "source": [
    "from scipy import stats\n",
    "from functools import partial\n",
    "from matplotlib import pyplot \n",
    "%matplotlib inline\n",
    "\n",
    "dist_continuous = [d for d in dir(stats) if isinstance(getattr(stats,d), stats.rv_continuous)]\n",
    "dist_discrete = [d for d in dir(stats) if isinstance(getattr(stats,d), stats.rv_discrete)]\n",
    "\n",
    "print \"CONTINUOUS\"\n",
    "print dist_continuous\n",
    "print \"\\nDISCRETE\"\n",
    "print dist_discrete\n",
    "\n",
    "def cumSum(L,y):\n",
    "    L.append(y+L[len(L)-1])\n",
    "    return L\n",
    "def cumE(L,y):\n",
    "    L.append(y/float(len(L)+1))\n",
    "    return L\n",
    "def cumSum2(L,y):\n",
    "    L.append(y**2+L[len(L)-1])\n",
    "    return L\n",
    "def cumVar(L,y,E):\n",
    "    i = len(L)\n",
    "    L.append(y/float(len(L)+1) - E[i]**2)\n",
    "    return L\n",
    "\n",
    "def sampleCumE(S):\n",
    "    sums = reduce(cumSum, S[1:N], [S[0]])\n",
    "    E = reduce(cumE, sums[1:N], [sums[0]])\n",
    "    return E\n",
    "\n",
    "def sampleCumVar(S,E):\n",
    "    sums2 = reduce(cumSum2, S[1:N], [S[0]**2])\n",
    "    V = reduce(partial(cumVar,E=E), sums2[1:N], [sums2[0]-E[0]**2])\n",
    "    return V\n",
    "\n",
    "def plotSampleEVar(S, ax=None):\n",
    "    E = sampleCumE(S)\n",
    "    Var = sampleCumVar(S,E)\n",
    "    N = len(E)\n",
    "    fig = pyplot.figure()\n",
    "    if not ax:\n",
    "        ax = fig.add_subplot(1,1,1)\n",
    "    ax.set_xscale('log')\n",
    "    ax.plot(E)\n",
    "    ax.plot(Var, 'r')\n",
    "    ax.set_title(\"E[X] = {0}, Var[X]={1}\".format(E[N-1], Var[N-1]))\n",
    "  \n",
    "def plotFunction(t, f, domain, ax=None):\n",
    "    raynge = [f(k) for k in domain]\n",
    "    if not ax:\n",
    "        fig = pyplot.figure()\n",
    "        ax = fig.add_subplot(1,1,1)\n",
    "    ax.plot(domain,raynge)\n",
    "    ax.set_title(t)\n",
    "\n",
    "def plotPMF(name, rv, l, r, ax=None):\n",
    "    t = name + \" Probability Mass Function\"\n",
    "    plotFunction(t, rv.pmf, range(l,r+1), ax)\n",
    "    \n",
    "def plotPDF(name, rv, start = -10, stop = 10, delta = 1, ax = None):\n",
    "    t = name + \" Probability Density Function\"\n",
    "    steps = (stop-start)/float(delta) + 1\n",
    "    domain = [(start + i*delta) for i in range(0,int(steps))] \n",
    "    plotFunction(t,rv.pdf, domain, ax)\n",
    "    \n",
    "def plotCDF(name, rv, l, r, delta=None, ax=None):\n",
    "    t = name + \" Cummulative Distribution Function\"\n",
    "    if delta:\n",
    "        steps = (r-l)/float(delta) + 1\n",
    "        domain = [(l + i*delta) for i in range(0,int(steps))] \n",
    "        plotFunction(t,rv.cdf, domain, ax) \n",
    "    else: plotFunction(t, rv.cdf, range(l,r+1), ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Discrete Random Variables</h2>\n",
    "<h3>Bernoulli Random Variable</h3>\n",
    "Given an experiment with two possible outcomes, *S* and *F* (thought of as success and failure), with probability *p* for *S* and *1-p* for *F*, the **Bernoulli** random variable *X* is \n",
    "\n",
    "$$X =  \\left\\{ \\begin{eqnarray} 1, \\quad \\mbox{if} & S \\cr\n",
    "0, \\quad \\mbox{if} & F\n",
    "\\end{eqnarray} \\right.$$\n",
    "\n",
    "It's PMF is given by \n",
    "\n",
    "$$p_X\\left(k\\right) = \\left\\{ \\begin{eqnarray} p, \\quad \\mbox{if} & k=1 \\cr\n",
    "1-p \\quad \\mbox{if} & k=0\n",
    "\\end{eqnarray} \\right.$$\n",
    "\n",
    "Related quantities\n",
    "\n",
    "$$E\\left[X\\right] = p$$\n",
    "\n",
    "$$var\\left(X\\right) = p\\left(1-p\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=0.25, Var[X]=0.1875\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import bernoulli\n",
    "p = .25\n",
    "rv = bernoulli(p)\n",
    "N=1000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "plotSampleEVar(rv.rvs(size=N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Binomial Random Variable</h3>\n",
    "Given an experiment with two possible outcomes, *S* and *F*, with probability *p* for *S* and *1-p* for *F*, that is repeated *n* times, the **Binomial** random variable is the number of experiments resulting in an *S* outcome denoted by *k*, i.e. the number of successes in *n* trials. It's PMF is given by\n",
    "\n",
    "$$p_X\\left(k\\right) = {n \\choose k} p^k \\left(1-p\\right)^{n-k} \\quad k=0,1,\\ldots,n$$\n",
    "\n",
    "Related values\n",
    "\n",
    "$$E\\left[X\\right] = np$$\n",
    "\n",
    "$$var\\left(X\\right) = np\\left(1-p\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=2.5, Var[X]=1.875\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import binom\n",
    "n = 10\n",
    "k = 4\n",
    "rv = binom(n,p)\n",
    "N = 1000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPMF(\"Binomial\", rv, 1, 10, ax=axarr[1])\n",
    "plotCDF(\"Binomial\", rv, 1, 10, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Geometric Random Variable</h3>\n",
    "Given an experiment with two possible outcomes, *S* and *F*, with probability *p* for *S* and *1-p* for *F*, that is repeated until an *S* outcome is observed, the **Geometric** random variable is the number of experiments required including the *S*, i.e. the number of trials required to reach the first success. It's PMF is given by\n",
    "\n",
    "$$p_X\\left(k\\right) = p \\left(1-p\\right)^{k-1}$$\n",
    "\n",
    "Related Values:\n",
    "\n",
    "$$E\\left[X\\right] = \\frac{1}{p}$$\n",
    "\n",
    "$$var\\left(X\\right) = \\frac{1-p}{p^2}$$\n",
    "\n",
    "CDF:\n",
    "$$F_X\\left(k\\right) = 1-(1-p)^n$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=4.0, Var[X]=12.0\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import geom\n",
    "p = 0.25\n",
    "rv = geom(p)\n",
    "N = 2000\n",
    "\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPMF(\"Geometric\", rv, 1, 20, ax=axarr[1])\n",
    "plotCDF(\"Geometric\", rv, 1, 20, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Poisson Random Variable</h3>\n",
    "The **Poisson** random variable is a random variable whose PMF is\n",
    "\n",
    "$$p_X\\left(k\\right) = e^{-\\lambda}\\frac{\\lambda^k}{k!}$$\n",
    "\n",
    "where $\\lambda\\hspace{1pt}$ is a positive parameter characterizing the PMF. The Poisson RV arrises as an **approximation to the Binomial RV**. Given the condition of *large n* and *small p* and $k\\ll n \\hspace{1pt}$ then letting $\\lambda = np \\hspace{1pt}$ in the Poisson PMF provides a good approximaiton to the Binomial PMF with parameters *n*, *k*, and *p* with the advantage of a simpler calculation.\n",
    "\n",
    "Related values:\n",
    "\n",
    "$$E\\left[X\\right] = var\\left(X\\right) = \\lambda$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=0.25, Var[X]=0.25\n"
     ]
    },
    {
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       "RERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERERE\n",
       "RERERERERERERERERERERERERERERCTn/j/KqAhM3G6VSwAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed19910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ed192d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import poisson\n",
    "lam = 0.25\n",
    "rv = poisson(lam)\n",
    "N = 5000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPMF(\"Poisson\", rv, 1, 5, ax=axarr[1])\n",
    "plotCDF(\"Poisson\", rv, 1, 5, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Discrete Uniform Random Variable</h3>\n",
    "A random variable $X$ is a **discrete** uniform random variable if it can take any of *n* values with equal probability, namely $frac{1}{n}\\hspace{1pt}$. Assume the values are indexed to (or are) integers in the range $\\left[a,b\\right]\\hspace{1pt}$ then the PMF of *X* is\n",
    "\n",
    "$$p_X\\left(k\\right) = \\left\\{ \\begin{eqnarray} \\frac{1}{b-a+1}&,& \\quad \\mbox{if} \\hspace{3pt} k=a,a+1,\\ldots,b \\cr\n",
    "0&,& \\quad\\mbox{otherwise} \\end{eqnarray}\\right.$$\n",
    "\n",
    "Related values:\n",
    "\n",
    "$$E\\left[X\\right] = \\frac{a+b}{2}$$\n",
    "\n",
    "$$var\\left(X\\right) = \\frac{\\left(b-a\\right)\\left(b-a+2\\right)}{12}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2>Continuous Random Variables</h2>\n",
    "<h3>Continuous Uniform Random Variable</h3>\n",
    "A random variable *X* is a **continous** uniform if it can take all values in the interval $\\left[a,b\\right]\\hspace{1pt}$ and the probability of *any two equal length intervals* is the same, so that the PDF is\n",
    "\n",
    "$$f_X\\left(x\\right) = \\left\\{ \\begin{eqnarray} \\frac{1}{b-a}&,& \\quad \\mbox{if} \\hspace{3pt} a\\le x \\le b \\cr\n",
    "0&,& \\quad\\mbox{otherwise} \\end{eqnarray}\\right.$$\n",
    "\n",
    "Related values:\n",
    "\n",
    "$$E\\left[X\\right]= \\frac{a+b}{2}$$\n",
    "\n",
    "$$var\\left(X\\right) = \\frac{\\left(b-a\\right)^2}{12}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=0.5, Var[X]=0.75\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import uniform\n",
    "a=-1\n",
    "b=2\n",
    "rv = uniform(loc=a,scale=(b-a))\n",
    "#This distribution is constant between loc and loc + scale.\n",
    "N = 1000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPDF(\"Uniform\", rv, -5, 5, .05, ax=axarr[1])\n",
    "plotCDF(\"Unfiform\", rv, -5, 5, .01, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Exponential Random Variable</h3>\n",
    "\n",
    "An exponential RV has PDF\n",
    "\n",
    "$$f_X\\left(x\\right) = \\left\\{ \\begin{eqnarray} \\lambda e^{-\\lambda x}&,& \\quad \\mbox{if} \\hspace{3pt} x \\ge 0 \\cr\n",
    "0&,& \\quad\\mbox{otherwise} \\end{eqnarray}\\right.$$ \n",
    "\n",
    "Note that we require $\\lambda \\gt 0$\n",
    "\n",
    "Related Values:\n",
    "\n",
    "$$E\\left[X\\right] = \\frac{1}{\\lambda}$$\n",
    "\n",
    "$$var\\left(X\\right) = \\frac{1}{\\lambda^2}$$\n",
    "\n",
    "CDF:\n",
    "$$F_X\\left(x\\right)=\\int_0^{\\infty}\\lambda\\exp^{-\\lambda t}dt=1-\\exp^{-\\lambda t}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=4.0, Var[X]=16.0\n"
     ]
    },
    {
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       "yxQAAAAASUVORK5CYII=\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import expon\n",
    "#The scale parameter is equal to scale = 1.0 / lambda.\n",
    "lam = .25\n",
    "scale = 1.0 / lam\n",
    "rv = expon(scale=scale)\n",
    "N = 1000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPDF(\"Exponential\", rv, 0, 25, .5, ax=axarr[1])\n",
    "plotCDF(\"Exponential\", rv, 0, 25, .5, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h3>Normal Random Variable</h3>\n",
    "\n",
    "A continuous random variable *X* is **normal** or **Gaussian** if it has a PDF of the form\n",
    "\n",
    "$$f_X\\left(x\\right) = \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{\\left(x-\\mu\\right)^2/2\\sigma^2}$$\n",
    "\n",
    "Related Values\n",
    "\n",
    "$$E\\left[X\\right] = \\mu$$\n",
    "\n",
    "$$var\\left(X\\right) = \\sigma^2$$\n",
    "\n",
    "The RV *Y* is said to be a **standard normal** if it is normally distributed with zero mean and unit variance. Given any normally distributed random variable, *X* with mean $\\mu\\hspace{1pt}$ and variance $\\sigma^2\\hspace{1pt}$ and noting that *normality is preserved by linear transformations*, we can obtain a *standardized* form of *X* as follows\n",
    "\n",
    "$$Y = \\frac{X-\\mu}{\\sigma}$$\n",
    "\n",
    "$$E\\left[Y\\right] = \\frac{E\\left[X\\right]-\\mu}{\\sigma}=0$$\n",
    "\n",
    "$$var\\left(Y\\right) = \\frac{var\\left(X\\right)}{\\sigma^2}=1$$\n",
    "\n",
    "Finally, the CDF of *X* can be obtained from the standard normal CDF, $\\Phi\\left(y\\right)\\hspace{1pt}$, by \n",
    "\n",
    "$$P\\left(X\\le x\\right) = P\\left(\\frac{X-\\mu}{\\sigma}\\le \\frac{x-\\mu}{\\sigma}\\right)=P\\left(Y\\le \\frac{x-\\mu}{\\sigma}\\right) = \\Phi\\left(\\frac{x-\\mu}{\\sigma}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theoreticals: E[X]=1.0, Var[X]=9.0\n"
     ]
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "#The scale parameter is equal to scale = mu, loc = sigma = standard deviation\n",
    "mu = 1\n",
    "sig = 3\n",
    "rv = norm(loc=mu,scale=sig)\n",
    "N = 1000\n",
    "print \"Theoreticals: E[X]={0}, Var[X]={1}\".format(rv.mean(), rv.var())\n",
    "f,axarr = pyplot.subplots(1,3)\n",
    "f.subplots_adjust(right=2.5)\n",
    "f.subplots_adjust(top=1.5)\n",
    "plotSampleEVar(rv.rvs(size=N),axarr[0])\n",
    "plotPDF(\"Normal\", rv, mu-4*sig, mu+4*sig, .05, ax=axarr[1])\n",
    "plotCDF(\"Normal\", rv, mu-4*sig, mu+4*sig, .01, ax=axarr[2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
